en>Conquerist: Disambiguated: WDM → Wavelength-division multiplexing, SDH → Synchronous Digital Hierarchy

2013-10-04T17:13:26Z

Disambiguated: WDM → Wavelength-division multiplexing, SDH → Synchronous Digital Hierarchy

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{{for|Hua's identity in [[Jordan algebra]]s|Hua's identity (Jordan algebra)}}
In algebra, '''Hua's identity'''<ref>{{harvnb|Cohn|2003|loc=§9.1}}</ref>  states that for any elements ''a'', ''b'' in a [[division ring]],
:<math>a - (a^{-1} + (b^{-1} - a)^{-1})^{-1} = aba</math>
whenever <math>ab \ne 0, 1</math>. Replacing <math>b</math> with <math>-b^{-1}</math> gives another equivalent form of the identity:
:<math>(a+ab^{-1}a)^{-1} + (a+b)^{-1} =a^{-1}.</math>

An important application of the identity is a proof of '''Hua's theorem'''.<ref>{{harvnb|Cohn|2003|loc=Theorem 9.1.3}}</ref><ref>http://math.stackexchange.com/questions/161301/is-this-map-of-domains-a-jordan-homomorphism</ref> The theorem says that if <math>\sigma: K \to L</math> is a [[function (mathematics)|function]] between division rings and if <math>\sigma</math> satisfies:
:<math>\sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^{-1}) = \sigma(a)^{-1},</math>
then <math>\sigma</math> is either a [[Ring homomorphism|homomorphism]] or an antihomomorphism. The theorem is important because of the connection to the [[fundamental theorem of projective geometry]].

== Proof ==

: <math>(a - aba)(a^{-1} + (b^{-1} - a)^{-1}) = ab(b^{-1} - a)(a^{-1} + (b^{-1} - a)^{-1}) = 1.</math>

== References ==
{{reflist}}
* {{cite book | first=Paul M. | last=Cohn | authorlink=Paul Cohn | edition=Revised ed. of Algebra, 2nd | title=Further algebra and applications | year=2003 | location=London | publisher=[[Springer-Verlag]] | isbn=1-85233-667-6 | zbl=1006.00001 }}

[[Category:Theorems in algebra]]

{{algebra-stub}}

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en>Conquerist: Disambiguated: WDM → Wavelength-division multiplexing, SDH → Synchronous Digital Hierarchy